"It's a moment I'll cherish and hopefully I can do the coaches justice for giving me the start," said Weir.
Elsewhere, Euan Murray, who was absent for the victory over Ireland due to his Christian faith, returns at tighthead prop with Geoff Cross moving to the bench.
Ryan Wilson, the 23-year-old Glasgow Warriors No 8 who has led his club on three occasions this season and who was captain and man of the match in Scotland A's victory over England Saxons last month, is introduced to a match-day 23 for the first time.
Wilson is named instead of David Denton who sustained a knee injury when playing for Edinburgh Rugby against Scarlets at the weekend.
Johnson added: "Geoff (Cross) was a standout against Ireland and it's up to Euan, who fronted up big time against Italy, to raise the bar again.
"Wales will be going for their fifth successive away victory in the championship and that would be a record for them but our focus is on ensuring we win the contact zone, continue to improve and make it three wins on the bounce."
15 Stuart Hogg (Glasgow Warriors)
14 Sean Maitland (Glasgow Warriors)
13 Sean Lamont (Glasgow Warriors)
12 Matt Scott (Edinburgh Rugby)
11 Tim Visser (Edinburgh Rugby)
10 Duncan Weir (Glasgow Warriors)
9 Greig Laidlaw (Edinburgh Rugby)
1 Ryan Grant (Glasgow Warriors)
2 Ross Ford (Edinburgh Rugby)
3 Euan Murray (Worcester Warriors)
4 Richie Gray (Sale Sharks)
5 Jim Hamilton (Gloucester)
6 Robert Harley (Glasgow Warriors)
7 Kelly Brown (Saracens) captain
8 Johnnie Beattie (Montpellier)
16 Dougie Hall (Glasgow Warriors)
17 Moray Low (Glasgow Warriors)
18 Geoff Cross (Edinburgh Rugby)
19 Alastair Kellock (Glasgow Warriors)
20 Ryan Wilson (Glasgow Warriors)
21 Henry Pyrgos (Glasgow Warriors)
22 Ruaridh Jackson (Glasgow Warriors)
23 Max Evans (Castres)